You hav to rely completely on Egg Moves, TMs, and, MTs for this guy, unless you want to use Teleport. Thank goodness it has good enough Speed to allow for sweeping, cause it can't defend for s***. I think:

Psychic: STAB, good power, blah blah blah. You know by now what I have to say for moves, I don't think I have to explain some things.

Shock Wave: Takes care of those pesky Water types who seem to be everywhere.

2 Answers

Psychic(TM)-This is given, a STAB move on a sweeping Abra, who would've guessed.

shadow ball-Covers Ghost weakness

signal beam-covers dark weakness, assuming no occasional houndour. (I don't think he has to worry too much about bug types, seeing as most of the best bug Types are banned in LC)

Energy ball/ reflect/barrier-Energy ball can still cover those common water types, with higher power, and there's no need to worry about them having an ice type as well since ice isn't resistant to grass. Reflect and barrier boost his defense, your choice.

Here is a set I just made for my Abra:
Abra @Twisted Spoon
Adamant Nature
EVs: 252 Atk. 248 Spe 8 Sp. Def
Trait: Syncrinoise
Magic Coat
Knock Off
Psychic Terrain
Zen Headbutt
Magic Coat reflects any status conditions like Thunder Wave or Toxic back to the Opponent. It works only on the turn it is used. Good For toxic Stallers and beats bringing in a magic bounce user.
Knock off to knock off an opponent's item such as a life orb or leftovers so their moves cannot be boosted or so they cannot restore HP each turn. Watch out for HP draining moves and Annoying Aqua Ring.
Psychic Terrain Boosts the power of psychic type moves. So Zen headbutt can be more powerful. I chose Zen headbutt over psychic because that move is too mainstream. Also, Zen Headbutt has the chance to make the opponent Flinch.

Hope this helped! Saw there was only one answer since 8 years ago so this board needs some more answers :) Glad to help in advance. On a side note, this is a set for a supporter in Doubles LC. Not to be mixed for a bad set. I know it's attack stat is pretty bad, but It is enough to cause a foe to flinch and enough to knock off someone's item.